This section collects necessary preliminaries. Note that by Theorem REF , a necessary condition for \(\pi \)  to have non-zero Dirac cohomolgy is that \(\Lambda \)  is real in the sense of [1]}. That is, \(\Lambda \in i \mathfrak {t}_{f, 0}^* + \mathfrak {a}_{f, 0}^*\) . If not stated otherwise, we will assume that \(\Lambda \)  is real henceforth.
